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Full text of "Treatise On Analysis Vol-Ii"

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396 XV NORMED ALGEBRAS AND SPECTRAL THEORY Let Fffc denote the sum of the subspaces T(<pLik)(Rji) for 1 <£y g /, and let Fik denote the image of Fjfc under the invertible operator M(wtl) •- Af(wljk)\ M(w21) ••• Af(H>2fc)\ M(wkl) ••• M(wkk)] on Gk . Then F* is the sum of the Ftk (1 ^ / ^ &). (Consider the projector PF •) 11. THE SPECTRAL THEORY OF HILBERT (15.11.1) Let E be a separable Hilbert space and TV a normal continuous operator on E (15.4.1 1 ). If.fi/is the closure of the (commutative) subalgebra of &(E) generated by 1E, N and N*, then ((15.4.13) and (15.4.15)) s/ is a separable star algebra, and the mapping co : ji— >#(7V) is a homeomorphism of X(cC/) onto Sp(TV) c C (the spectrum of N with respect to J*?(E), that is to say the spectrum of N as defined in (11.1)). Hence it follows from the Gelfand- Neumark theorem (15.4.14) that the mapping /W ^ " * (/ o co) is a faithful representation of ^c(Sp(AO) on E, which can therefore be extended to a faithful representation of ^c(Sp(AO) on E (15.10.8). We denote this represen- tation by/W/(AO. This notation is justified by the fact that, if /and g are any two functions belonging to ^c(Sp(AO), then in the algebra & (E) we have =/(AO + g(N), (fg)(N) = (15.11.1.1) (15.11.1.2) (where lSp(JV) is the identity mapping f H» f of Sp(N)). The Gelfand-Neumark theorem also shows that, for every continuous function / on Sp(JV), we have (15.11.1.3) ||/(N)||=P(/(JV))= sup |/(C)| (cf. (15.11.8.1)). Moreover ((15.10.3) and (15.10.8)), there exists a continuous sesquilinear mapping (x, y) ^mx>y of E x E into the Banach space Mc(Sp(AT)) of complex measures on Sp(JV) such that my> x = mx y and (15.11.2) (/(JV) • x | y) = /(O dm, .(0be identified with LC(^*)» and